Solution of Linear Ordinary Differential Equations with Polynomial Coefficients

The question of the existence of a closed-form solution of a given linear Ordinary Differential Equations with polynomial coefficients is an interesting one. In solving such equations by the power series methods, we do not know in advance whether the solution is in a closed-form or not. In this paper, we show that we can answer this question with simple calculations, and then, we express the weak and strong aspects of this method by considering some examples.